Simple Method for Measuring Dispersion and Spectral Absorption of Si wafers for use in MEMS Applications Mohamed Nabil, and Diaa Khalil Senior Member IEEE
Mohamed Nabil, and Diaa Khalil Senior Member IEEE
Faculty of Engineering Ain Shams University Cairo, Egypt
[email protected] [email protected]
MEMS division Si-Ware Systems SWS Cairo, Egypt
[email protected] [email protected]
Abstract—This paper describes a simple method for measuring the refractive index and absorption coefficient of silicon wafers in the Infrared (IR) wavelength range using a fiber probe. The proposed technique is applied in the wavelength range 1 to 1.7 microns. Results are compared to published models and a good agreement is obtained. Keywords-silicon, photonics, Dispersion, beam splitter, optical MEMS.
I.
S
INTRODUCTION
ILICON is nearly the default material in electronic applications especially for integrated circuits fabrication. Recently Si wafers have been also targeted as a base for optical integrated circuits either in the form of integrated guided wave components, optical MEMS components or even as a laser source. One of these interesting applications of high potential is the use of a thin sheet of Si [1] or the Si / Air interface [2], [3] as a beam splitter in building an optical interferometer using MEMS technology. This allows having a monolithic integrated self-aligned optical interferometer fabricated in one lithography step using the Deep Reactive Ion Etching (DRIE) technology. In such application; part of the light beam travels inside the silicon and may suffer from dispersion effects due to the wavelength dependence of the silicon refractive index. This effect leads to phase error in the obtained interferogram and thus to a spread of the interferogram especially when a white light source with large spectrum is used. In Fourier transform spectrometers this effect has a strong impact on the interferometer resolution for a given mirror travel range. Propagation inside silicon leads also to attenuation of the light beam power due to absorption which is characterized by the absorption coefficient. Both the absorption coefficient and the refractive index are sensitive parameters that vary with wavelength, temperature, doping concentration and even wafer orientation. An accurate simple method for the
determination of the Si wafer group refractive index as well as spectral attenuation is thus of great importance for the development of MEMS FT spectrometers. The objective of this work is thus to develop a simple technique for the characterization of the Si wafer refractive index and spectral attenuation while it is in its wafer format. The paper is also focusing on the use of fiber probe in the characterization to allow for small areas measurement and thus non-homogeneity estimation. The paper will be presented as follows: In section II the refractive index measurement will be presented, in section III, the spectral attenuation measurement will be presented and finally in section IV, a brief conclusion will be given. II.
REFRACTIVE INDEX MEASUREMENT
Phase error in optical interferometers is essentially a function of the group refractive index of medium in which the light is propagating. For group refractive index measurement we used the transmission response of a Fabry-Perot resonator [4] which is formed by a silicon wafer sample assuming that the two faces of the sample will be acting as the interferometer end mirrors. For this purpose we used a silicon wafer highly polished from both sides. The wafer thickness is measured using a digital micrometer with accuracy better than 1 micron and the thickness is found to be 484 µm. The free spectral range (FSR) of a Fabry-Perot resonator is given by (1) FSR = co / (2 Nd ) , where co is the speed of light in free space, d is the length of the resonator, and N is the group refractive index of the medium between the two mirrors. By taking the Fourier transform of the resonator spectral response, N can be calculated from the first peak position where the Fourier transform pair in this case is given by
∞
∫−∞ f (k ) exp(− jkx)dk , (2) ∞ 1 f (k ) = f ( x) exp( jkx)dx 2π −∫∞ where k is the wave number or the spatial frequency and is related to the wavelength (λ) by k=2π/λ, and to the frequency (ν) by k=2πν/c. The free spectral range in k space is then given by (3) FSRk = π /(Nd ) .
-11
x 10
f ( x) =
Like the Fourier transform of periodic time domain signals, we note that equating the x value (xp) of the first peak of f(x) to 2π/FSRk, we get xp=2Nd, from which we calculate N. We used a white light source of high power in the required wavelength range coupled to a lensed AR-coated standard single mode fiber. The lensed end of the fiber is fixed in front of the sample as shown in Fig. 1. The output fiber is a multimode one and is connected to an optical spectrum analyzer (OSA). The input and output fibers are brought very close to the sample. The wavelength range of interest is divided into a number of sub-ranges of equal span. We made measurements for spans of 20, 10, and 5 nm. The measured group refractive index is compared to a reference model [5] for each span value. The 5 nm span which contains at least 8 resonance frequencies was found to be accurate enough. The optical spectrum analyzer was controlled by GPIB commands to analyze and send to the PC the spectrum of each sub-range individually. The resolution of the spectrum analyzer was set to 0.005 nm at a 5 nm span which is sufficiently smaller than the predicted maximum FSR of 0.3 nm (Fig. 2). The OSA data are stored on the PC then FFT is done on each sub-range to find the group refractive index at the center wavelength of the corresponding sub-range (Fig. 3) as explained before. Fig. 4 shows the measured group refractive index for the three span values together with the values reported in the reference model.
Power (x10-11 W)
1.6 1.4 1.2 1 0.8 0.6
1.498
1.499 1.5 1.501 W avelength (µm)
1.502 -6 x 10
Figure 2. Spectrum of transmitted light in the output fiber in 5 nm span window centered on 1550 nm.
Using a curve fitting technique, a fifth order polynomial was found to fit the measured group refractive index of 5 nm span (Fig. 5) and is given as (4) N (λ ) = c1 λ5 + c 2 λ 4 + c 3 λ3 + c 4 λ 2 + c 5 λ + c 6 , -15 -11 -8 where c1=1.802·10 , c2=-1.009·10 , c3=2.018·10 , c4=1.53·10-5, c5=0, c6=7.339, and λ is in microns. The constant c5 was forced to be zero in the fitting tool to avoid problems when integrating N(λ) to get n(λ) as will be seen latter. The relation between group refractive index (N) and absolute refractive index (n) is given by (5) N (λ ) = n − λ ( dn / dλ ) , from which we can see why N decreases as span (dλ) increases as noted before in our measurements (Fig. 4), where dn/dλ is always negative. The refractive index can then be calculated by (6) n(λ ) = −λ ∫ ( N / λ2 ) dλ + cλ , where c is a constant. Substituting N from (4) in (6), we get (7) n(λ ) = c 6 − c1λ5 / 4 − c 2 λ 4 / 3 − c3 λ3 / 2 − c 4 λ2 + cλ , where λ is in microns. -12
2
x 10
Si wafer
Optical fibers
a.u.
1.5
1
0.5
Figure 1. Si wafer is inserted between the aligned input and output fibers just touching the tips.
0 -0.1
-0.05
0 2Nd (m)
0.05
0.1
Figure 3. Fourier transform of measured spectrum in Fig. 2 by using FFT, where the position of the peak is used to compute the group refractive index at the center wavelength.
3.9 Ref. model 20 nm span 10 nm span 5 nm span
3.8 3.75 3.7 3.65 3.6 3.55 3.5 1100
1200
1300 1400 1500 W avelength (nm)
1600
1700
Figure 4. Measured group refractive index at three different span values together with the one calculated from the reference published model.
Now, we need to know n at a single wavelength to determine the value of the constant c. For this purpose we used the technique of lateral shift suffered by a beam when passing through a plate at a fixed wavelength (Fig. 6). It can be shown that the refraction angle θ1 is related to the shift x by (8) where all symbols are defined in Fig. 6. 3.9 Measured Fitting
Group refractive index
3.85
We can now calculate the value of the constant c using the measured value of n at λ=1550 nm from (7). We got c=-0.0119. The measured spectral refractive index and the one of the reference model are shown in Fig. 8.
3.8 3.75 3.7
0.7
3.65
0.6
3.6
0.5
3.55 1100
1200
1300 1400 1500 W aelength (nm)
1600
1700
Figure 5. Measured group refractive index using 5 nm span and the fifth degree fitting polynomial.
Measured Fitting
0.4 sinθ
Group refractive index
3.85
sin θ − ( x / d ) (8) cos θ We measured x at different θ and calculated θ1 from (8) then fitted the relation between sinθ and sinθ1 to a straight line passing by the origin (Fig. 7). The slope of this line gives the value of n at the used wavelength which was 1550 nm. We got n=3.46 with less than 0.5% error if compared to the reference model. A more accurate but complex methods may be used for this step like the method of minimum deviation. It should be noted that the setup had some modifications for this experiment. The output fiber is replaced by a standard single mode one so as to catch the first order transmitted beam only, it is easy to see that the separation (s) between maxima of output beams is given by (9), and assuming n=3.5 we get s=24 µm which is much larger than the core diameter of the standard single mode fiber of about 10 µm. The sample is fixed on a rotary stage to measure the angle. This rotary stage is fixed on an x-ypositioner to align the axis of rotation with the sample axis, so that the sample rotates about its axis keeping the separation with input and output fibers almost the same. The x-y positioner is fixed on a lifting stage to allow aligning the input and output fibers before inserting the sample. The input fiber is connected to 1550 nm laser source while the output one is fixed on a micrometer and is connected to a detector. (9) s = d sin( 2θ ) / n 2 − sin 2 θ
θ1 = tan −1
0.3 0.2 0.1 0 0
0.05
0.1 sinθ 1
0.15
0.2
Figure 7. The sine of the incidence angle versus the sine of the measured refraction angle and an origin-passing fitting line whose slope in the required refractive index.
Figure 6. The lateral shift suffered by a beam when passing through a plate as a method to measure the refractive index at a single tone.
-10
3.56
9 Measured Ref. model
x 10
P
1
8
P
3.54
2
3.52
Power (x10-10 W)
Refractive index
7
3.5 3.48
6 5 4 3 2
3.46 1 3.44 1100
1200
1300 1400 1500 W avelength (nm)
1600
Figure 8. Measured refractive index together with a reference model.
III.
ABSORPTION COEFFICIENT MEASUREMENT
For the measurement of absorption coefficient, we used the same setup of group refractive index measurement. The OSA was configured to display the wavelength range 1000 nm to 1600 nm with a resolution 0.7 nm. The measured spectrum was recorded using a PC with the sample inserted (P2) and without it (P1) and we repeated these two measurements 20 times then took the average to work with (Fig. 9). We accounted for the multiple reflections inside the sample by considering it a lossy Fabry-Perot filter. The transmission response of a lossy Fabry-Perot filter (Tfp) is given by T2 , (10) T fp = αd e (1 − R e −αd ) 2 + 4 R sin 2 (nkd ) where T and R are the interface transmission and reflection coefficients respectively, R=(n-1)2/(n+1)2, T=1-R, n is the refractive index of the medium which is silicon in our case, α is the absorption coefficient, d is the thickness of the sample, and k is the wave number. In our measurements we use high resolution that eliminates the oscillating behavior of the Fabry-Perot and use an averaging technique on our results to remove the associated ripples. Thus taking the average of (10) over wave number we get T 2 e −αd . (11) T fpav = 1 − R 2 e − 2αd To include the effect of beam divergence, we considered the emerging beam from the lensed AR-coated single mode fiber to be a Gaussian beam of Rayleigh distance zo=150 µm as specified by the manufacturer at the wavelength of 1550 nm. We also assumed that the power coupled to the output fiber is the whole power falling on its core area with radius ρm=50 µm. The electric field of a Gaussian beam is given by ρ2
E(ρ , z) =
2 P − w2 − j kz p − tan e e yπw 2
−1 z d
zo
+k
ρ2 2 R
,
0 1000
1700
(12)
where P is the beam power, y is the medium admittance, w is the beam waist calculated by
1100
1200 1300 1400 W avelength (nm)
1500
1600
Figure 9. The average of 20 times measured spectrum in two cases; one without the sample (P1) and another with it (P2).
z 2 (13) w = wo 1 + d , z o k is the wave number, zp is the equivalent propagation distance calculated by (14) z p = ∑ ni z i , medium i
zd is the equivalent divergence distance and is calculated by (15) z d = ∑ z i / ni , medium i
and R is the phase front radius of curvature calculated by z 2 (16) R = z d 1 + o . z d The minimum beam waist wo is dependent on the wavelength λ and is related to the Rayleigh distance zo by
wo =
zo λ / π .
(17)
The concept of equivalent propagation and equivalent divergence distances is proposed to account for a Gaussian beam travelling in different successive media. The equivalent propagation distance is the summation of the product of the refractive index times the physical distance traveled in each medium (14), while the equivalent divergence distance is the summation of the product of the physical distance divided by the refractive index of each medium (15). Now, the power coupled to the output fiber (P') can be calculated by P ′ = ∫∫ y E dA = 2
2ρ − 2m P ′ = P 1 − e w 2
2π ρ m
∫ ∫yE
2
ρdρdφ
0 0
,
(18)
and the measured powers P1 and P2 are then given by 2ρ − 2m P1 = Po 1 − e w1 2
,
(19)
2ρ − m2 P2 = PoT fpav 1 − e w2 2
IV.
,
(20)
where w1 is the beam waist at the output fiber when the sample is removed and w2 is the beam waist at the output fiber when the sample is inserted. We then used (13) to calculate w1 and w2 where we substituted zd1 from (21) z d 1 = d − z o + 50 µm for zd in calculating w1 where we assumed a 50 µm thickness of the air gaps between fibers and sample surfaces, and substituted zd2 from (22) z d 2 = d / n − z o + 50 µm in calculating w2 where n is the silicon refractive index measured before. The relation between P2 and P1 is now given by 2 T 2 e −αd P1 1 − exp(−2 ρ m2 / w22 ) T1 e −αd P1 (23) = P2 = 2 − 2α d 2 2 2 − 2α d 1− R e 1 − exp(−2 ρ m / w1 ) 1 − R e with 1 − exp(−2 ρ m2 / w22 ) , 2 T1 = T 2 2 2 1 − exp(−2 ρ m / w1 )
REFERENCES [1]
[2]
[3]
[4]
and so α can be calculated from −1 2 2 T1 2 P1 1 2 T1 P1 (24) 2 . + 4R α = ln 2 R − + d P P 2 2 Fig. 10 shows the calculated α together with a reference one [6]. It is noted that the absorption coefficient is almost zero after 1100 nm while it is increasing rapidly in the direction of lower wavelength. We note also that for some values α appears as if it is lower than zero. This is due to the incomplete correction of the diffraction effect as the diffraction in Si is lower than that in air. A more complicated analysis including the multiple reflections between output fiber tip and sample surface as well as the divergence effect inside the sample on the Fabry-Perot transmission response may be needed for better estimation of the absorption coefficient. 60 Measured Ref. Model
Absorption coefficient (cm-1)
50 40 30 20 10 0 -10 1
1.1
1.2 1.3 1.4 W avelength (µm)
1.5
1.6
Figure 10. Measured absorption coefficient together with a reference model.
CONCLUSION
The group refractive index of a silicon wafer in the near IR is measured with high accuracy by a simple method using a fiber optic probe. The absolute value of the refractive index is then derived using the lateral shift of a single wavelength laser beam when passing through the wafer. The absorption coefficient of silicon was also measured in the near IR using the same fiber probe. Obtained results are in good agreement with published models. This technique allows the rapid in-site optical characterization of the Si wafers in addition to its possible direct extension for non-homogeneity estimation.
[5]
[6] [7]
[8]
Ming C. Wu, Olav Solgaard, and Joseph E. Ford, “Optical MEMS for Lightwave Communication,” J. Lightw. Technol., vol. 24, no. 12, pp. 4443–4445, Dec. 2006. Diaa Khalil, Hitham Omran, Mostafa Medhat, and Bassam Saadany, “Miniaturized tunable integrated Mach-Zehnder MEMS interferometer for spectrometer applications,” Invited talk at SPIE Photonics West in San Francisco, California, USA, 23-28 January 2010. Haitham Omran, Mostafa Medhat, Bassam saadany and Diaa Khalil, “Mach-Zehnder MEMS Interferometer with Two Si/Air Beam Splitters,” IDT09, Riyadh, Kingdom of Saudi Arabia, 8-11 November, 2009. Bahaa E. A. Saleh, and Malvin Carl Teich, “Fundamentals of Photonics,” John Wiley & Sons, Inc., ch. 3, ch. 5, 1991. Bradley J. Frey, Douglas B. Leviton, and Timothy J. Madison, “Temperature-dependent refractive index of silicon and germanium,” NASA Goddard space flight center. “Optical Properties of Silicon,” Virginia Semiconductor, Inc. Tarek A. Al-Saeed and Diaa Khalil, “Dispersion compensation in moving optical wedge Fourier Transform spectrometer,” Journal of Applied Optics, vol. 48, no. 20, pp. 3979-3987, 10 July 2009. Diaa Khalil, “The Si Optical Bench Technology: Challenges and Opportunities,” Invited talk at The 6th International Symposium on “High-capacity Optical Networks & Enabling Technologies” (HONET’ 09), Alexandria, Egypt, Dec. 2009.
